Modeling the Economic Benefits of Supplemental Irrigation
Marcelo Torres
Department of Economics
University of Brasília, Brazil
Richard Howitt
University of California, Davis, USA
Lineu Rodrigues
Embrapa Cerrados, Brazil
RESUMO: Modelos hidroeconômicos existentes na literatura são geralmente aplicados a
áreas áridas ou semi-áridas do mundo onde precipiração é muito baixa e as culturas
agrícolas são 100% irrigadas. Nesses modelos, a água disponível para os agricultores é
resultado da agregação dos volumes de água de superfície e subterrânea em reservatórios
contruídos ou naturais como rios, lagos e aquíferos. Nesse modelos, precipitação é
implicitamente considerada como uma das fontes de água somente via seus efeitos nos
volumes dos reservatórios. Contudo, na maioria das áreas agricultáveis do mundo, uma
significante parcela da água aplicada pelos agricultores vem de precipitação que cai
diretamente sobre as culturas e os sistemas de irrigação são usados como fontes
supplementares de água no caso de secas. Neste caso, precipitação e água dos reservatórios
não são perfeitamente substituíveis. Ao usar dados primários referentes à atividade agrícola
de uma subbacia hidrográfica no centro-oeste brasileiro onde precipitação e o uso
suplementar de água ocorrem, esse artigo demonstra que modelos hidroeconômicos padrão
tendem a subestimar os impactos das secas sobre a atividade agrícola. Isso é explorado
através da construção de dois modelos alternativos de produção agrícola acoplados a um
modelo hidrológico, onde o objetivo é a maximização da receita líquida regional. No
primeiro modelo de produção, a substituição entre precipitação e água de superfície estocada
em reservatórios é explicitamente formalizada e essas duas fontes de água são consideradas
separadamente mas interconectadas. No segundo, elas são simplesmente agregadas em uma
fonte única de água. Ambos modelos são utilizados para simulações de efeitos de secas
sobre o uso do solo e da água, preços sombra da água e a renda agrícola. Resultados
demonstram que no primeiro modelo não somente os retornos esperados para os agricultores
são consistentemente menores como, talvez mais importante, a variabilidade dos retornos é
maior.
Palavras-chave: microeconomia aplicada, recursos hídricos, renda agrícola, modelos
hidroeconômicos.
Área ANPEC: 11.
JEL: Q25, Q18, Q15.
ABSTRACT: Standard hydroeconomic policy models are usually applied to areas in the
world where precipitation is very low and crops are fully irrigated. As such, these models
pool the total annual stored precipitation plus other water supplies and assume that this total
water supply can be allocated by time and place. This water pooling approach treats
seasonal precipitation and irrigation water as fully substitutable. In many irrigated areas,
however, a significant amount of water used by crops comes from seasonal precipitation. In
fact the majority of the global irrigation systems are supplemental to seasonal rainfall and, in
1
this context, precipitation and water stored in reservoirs are not fully substitutable. By using
primary data from a sub watershed in Brazil, where precipitation and supplemental water
use occur, this paper shows that standard models are likely to under-estimate the predicted
drought impacts. The paper investigates this issue by setting up two alternative economic
models of regional net-revenue maximization based on a calibrated economic programming
model integrated with a mass balance hydrologic model. In one model the substitution
between precipitation and irrigation water is explicitly formalized and in the other
precipitation and irrigation water are aggregated together in a single water supply. The
models are used to estimate and predict the short-run impacts of precipitation cuts on
irrigation reservoir levels and the impacts of lower irrigation water supply on farmers’
agricultural income. In comparing the results, we find that the expected returns to farmers
are lower under the former model and the variability of returns is higher.
Keywords: Applied microeconomics, water resources, agricultural income, supplementary
irrigation, positive mathematical programming, hydroeconomic model.
INTRODUCTION
Farmers are the dominant water users worldwide and compete for water resources which are
heterogeneously distributed across time and space. Changes in water availability may affect
agricultural income, productivity and cropping strategies and have potential environmental
effects for the hydrologic system as a whole. Therefore, an accurate evaluation of how
farmers may react to different water resource policies or environmental scenarios (e.g.:
temperature, water supply, precipitation regimes etc) is important for policy makers and can
help to shed light on ways to increase farmers’ income and to alleviate poverty in many
parts of the rural world. This paper questions whether the simpler hydroeconomic models
that treat seasonal precipitation as part of the total irrigation water supply are sufficiently
accurate for policy analysis in regions where irrigation is supplemental to seasonal
precipitation.
Empirical modeling of the effects of alternative water allocation regimes on
agricultural communities and farmers’ reactions is complex and by definition
interdisciplinary. Several surveys of studies that develop models involving economics,
hydrology, ecology and agronomy have been published (Harou et al. 2009 and Booker et al
(2012)). Although all the studies attempt to integrate different disciplines, each one focuses
on a particular issue. For example, Rosegrant et al. (2000) , Cai et al. (2003) focus on
salinity and water availability for irrigation; Cai (2008) on the optimal strategies for water
allocation among competing sectors and Harou and Lund (2008) on water pricing, irrigation
and institutional constraints; Loucks (2006) focuses on the integration of economics and
ecology and Guan and Hubacek (2007) on the relationship between economic activity and
water quality; and finally Krol et al. (2006), Medellín-Azuara et al. (2008, 2010) and Harou
et al. (2006) focus on the estimation of drought and climate change impacts on water
availability for agriculture.
One of the aspects these studies have in common is that their models are applied to
arid or semi-arid regions of the world where rainfall during the growing season is minimal
or absent. Water available for irrigation in an arid or semi-arid region predominantly comes
from out of season precipitation, or is imported from regions where precipitation occurs
through a system of man-made channels, rivers, groundwater aquifers. For example, water
from Northern California and the Sierras is used to irrigate Central Valley and Southern
California regions. Similar transfers are found in Spain, Chile, Israel and Australia.
2
In many areas of the world, however, precipitation occurs during the growing season,
and a significant amount of water used for crop production comes from precipitation that
falls directly onto the crops. Rost et al (2009) in figure 3 show regions of the world where
there is a significant increase in net primary productivity due to irrigation. The largest areas
under irrigation are located in monsoon climates and Central American regions where
irrigation systems are supplemental to seasonal rainfall. During the rainy season in these
areas, it is common for farmers to experience several days without rainfall. Water storage in
surface reservoirs or underground aquifers provides a supply of irrigation water in the event
of a drought. When precipitation is lower than expected, farmers will, ceteris-paribus, react
by taking more water from surface or groundwater sources to irrigate their crops. That is,
precipitation and surface (or groundwater) water are substitute agricultural inputs.
Our point is that the predicted drought impacts from models that ignore the restrictions
implicit in precipitation and the surface (or groundwater) substitution relationship for
seasonal rainfall are likely to under estimate the severity of drought. For instance, if the
impacts of a reduction in precipitation on farmers' income are estimated considering only the
changes in the volume of the reservoirs or other water bodies, the impacts may appear to be
minimal if the amount of water in the reservoir is not a binding constraint on deliveries. If
however, the model allows for the fact that a reduction in precipitation may induce famers to
replace rainfall with water from reservoirs, the impacts of drought on income may be
significant even if the reservoirs´ volume is not binding, since access to irrigation water
from the reservoirs is costly due to water fees, pumping costs, and system maintenance
costs.
One way to deal with this issue is to include precipitation as an explicit input in the
production system as in Maneta et al. 2009 and Torres et al. 2012. Although these two
studies have made important contributions to the literature, gaps remain. In particular, the
integration of uncertain rainfall with irrigation supplies in a formal production function. The
models used in this study build and improve on these earlier models in that the calibration of
water use is now exact, and the stochastic nature of precipitation and its effect on income
risk is explicitly modeled. In this context, this paper measures the importance of correctly
specifying precipitation by setting up two alternative simulation models of regional
agricultural production. One where the substitution between precipitation and surface water
is explicitly formalized and another where precipitation and irrigation water are lumped
together in a single water supply quantity, as is generally done in the models applied to arid
and semi-arid regions cited above.
Both models are integrated with a hydrological model which estimates the amount of
water available for irrigation and precipitation in time and space. The agricultural
production models and the hydrological model are externally coupled and the resulting
hydroeconomic models are used to estimate and predict the short-run impacts of
precipitation cuts on irrigation reservoir levels and the impacts of lower irrigation water
supply on farmers’ agricultural income. As in Maneta et al., 2009, the focus is on the Buriti
Vermelho river sub-basin, situated in the Federal District, near Brasília. The database is
based on primary data collected from interviews performed during the dry and wet seasons
of the 2007/2008 agricultural year. More details are provided throughout the paper sections
below.
METHODOLOGY
In this paper, we develop an agricultural production model (hereinafter termed the economic
model) using a mathematical programming approach and apply it, using primary data from
the farm community of Buriti Vermelho located in a sub-watershed in the São Francisco
3
River Basin in Brazil. The model is calibrated to reflect the coexistence of irrigated and
rainfed agriculture in the region. Moreover, it takes into account seasonal precipitation
levels as one of the arguments of the crop specific multi-input, multi-output production
functions. In this model, water is introduced through two sources: from the surface water
storage system and from seasonal precipitation that falls directly onto the crops. The model
is specified so that farmers adjust their product mix, production technology, area under
cultivation, and water use in response to economic and physical changes. Specifically we
model the changes in relative input and product prices, the availability of surface water for
irrigation, for different levels of precipitation. The amount of water available to farmers and
the relationship between precipitation and the level of water in reservoirs is estimated with a
hydrologic model that follows a water balance approach and yields water availability
estimates on a monthly basis at the farm level.
Economic Model
The economic model is based on a class of models called Positive Mathematical
Programming or PMP, (Howitt 1995) and largely used in applied economics (House, 1987;
Howitt and Gardner, 1986; Kasnakoglu and Bauer, 1988; Lance and Miller, 1998;
Chatterjee et al, 1998; Paris and Howitt, 1998; Maneta et al 2009, Medellín-Azuara et al.
(2010), Torres et al. 2012, Howitt et al. 2012).
Each farmer g’s goal in a given agricultural year is to choose Xhi and Xhj, the amounts
of input h to be applied on rainfed crop i and on irrigated crop j respectively, in order to
maximize net agricultural income. More formally, the farmer´s problem is
max net [ pi qir ( X hi )  p j q irj ( X hj ) 
X hi , X hj
i, j
 ( phi X hi  phj X hj )  ( i e X
i
land i
 j X land j
  je
)]
(1)
h
The first two terms inside the brackets represent gross income where pi and pj are the unit
selling prices of rainfed crop i and irrigated crop j faced by farmer g. Rainfed and irrigated
crops are produced respectively with a rainfed production function qir ( X hi ) and an irrigated
production function qirj ( X hj ) . The superscript r denotes rainfed and ir, irrigated. Both
production functions are discussed in more detail below. The third term in equation (1)
represents total cash costs with rainfed and irrigated crops, in which ph is the price of input
h. The inputs considered are land (land), applied water (aw), hired labor (lb), family labor
(flabor) and materials (mat). This last category of inputs is an aggregate cost that includes
the expenditures on machinery, fertilizers, pesticides, seeds and any other input used in the
production of irrigated and rainfed crops. The last two terms, in parenthesis, represent the
implicit cost of land used in rainfed and irrigated crops. Both terms have an exponential
functional form, with parameters  i ,  i ,  j and  j and capture costs related to the farmer
g’s land allocation process that are not directly observed by the researcher. Such nonlinearity in costs may arise, for example, from managerial constraints, heterogeneity in land
quality and spatially non-uniform access to water on farms.1
1
For a more detailled discution on the functional form and the definition of the implicit land cost term please
see Medellín-Azzuara 2010.
4
The irrigated and rainfed production functions follow a CES (Constant Elasticity of
Substitution) functional form. More formally, for rainfed crops, production is represented by
qir

 Ai Precipi 


 bhi X hi 
h


i

.
(2)
Where Ai , bhi ,  and  i are parameters. When producing rainfed crops farmers can use all
inputs except surface water. So h in (2) refers to land, labor, family labor and materials only.
 1
The parameter  is defined as
, in which σ is the elasticity of input substitution. εi is

the parameter associated with returns to scale. If equals 1, greater than 1 or smaller than 1,
we have constant returns, decreasing returns or increasing returns respectively.
Pa
Precipi  e , where Pie is the precipitation that is expected to fall onto crop i and Pi a is the
Pi
actual amount of precipitation that falls onto crop i. Precipi is a measure of the realized
annual water stress in rainfed crop production. We note that the dryland model is calibrated
to data on the expected rainfall in the region, which is an exogenous parameter that the
farmer has to assess, and thus is not directly part of the farmer’s economic allocation
decisions, though as an external parameter, it does modify the input allocation first order
conditions. As such, the levels of inputs such as fertilizer and labor must also reflect the
farmer’s level of risk aversion, and so we do not add an explicit risk aversion term in the
objective function, as it is already implicit in the individual famer data used to calibrate the
production function and the crop specific land cost function
For irrigated crops (superscript ir) production is represented by
j


 
 .
qirj  Aj   bhj X hj
 h

(3)
Where the arguments and parameters are defined as above with the difference that famers
may use applied water (aw) to irrigate crop j. Applied water used on crop j by farmer g
(Xawj) is defined as the sum of the amount of water used from surface water (sw) on crop j by
farmer g (Xswj) and actual precipitation ( X pj ). That is,
X awj  X swj  X pj
(4)
Note that equation (4) implies that surface water based irrigation and seasonal precipitation
are perfect substitutes, and that the quantity of irrigation water applied in any period and
location is a function of the precipitation. In the irrigation production function, total water
use is now a control variable and is subject to the same set of first order conditions and
substitution conditions as the other inputs. A comparison of equations (2) and (3) shows that
in the dryland model stochastic rainfall modifies the scale parameter Ai, but in the irrigated
production function total water use is a fully controllable input.
An important distinction of this paper is that within the irrigation production function
specification there are two different and important ways of defining how precipitation enters
into the crop water supply. We will term these two models the “precipitation model” and the
“model with pooled constraints”. The key difference is that while irrigation water is an
allocatable resource in both time and location, precipitation is not allocatable in either time
5
or location, but falls evenly over all crop areas and varies stochastically within the growing
season. In the precipitation model the farmer is able to allocate only their irrigation water
within the seasonal constraints in response to changing precipitation. In the pooled model
the monthly precipitation is added to the irrigation supplies and the total water supply is
allocated optimally over crops and time. This widely used specification of pooled water
supplies allows a greater level of water substitution than the farmer actually faces, and thus
under-estimates the impacts of drought on the expected returns and variability of farm
income.
Before we proceed, a brief note on the input prices. The prices of land and hired
labor are unit prices (per hectare and man-hour respectively) paid by each farmer. For
materials, prices are not modeled as separate from quantities, so the expenditures (unitary
price paid times input quantity used) with each input in the materials category are summed
up by crop. The price of applied water ( X awj ) is the average cost with surface water
constructed as the sum of the costs of hired labor used in irrigation, pumping electricity, and
irrigation capital divided by the total amount of surface water used by farmer g on irrigated
crop j.
Parameter Estimation of the Production and Implicit Cost of Land Functions
The parameters of the production function and the land implicit cost terms are derived
analytically based on a system of equations that result from the first order conditions for
net-revenue maximization (FOC’s) and are subject to assumptions on the returns to scale
and the elasticity of input substitution and the land supply elasticity. The FOC’s state that
farmers maximize their net revenue by choosing an amount of input applied to crops i and j
such that the value of the input marginal product equals its marginal cost. This marginal
cost is composed of the sum of the market input price, if the input is traded in a market, and
the shadow price in the case of inputs with limited binding supply. Land, family labor, and
applied water are defined as subject to maximum supply constraints.
In the case of the land input, besides the market price and the limiting supply shadow
price, the PMP approach considers a third component of the marginal cost of land. This
component is the value of the Lagrange multiplier associated with a calibration constraint
that restricts the maximum amount of land that can be applied to a given crop to be equal to
the amount actually allocated in the base year. This Lagrange multiplier measures the
marginal implicit cost of land and is crop and farmer specific. All shadow prices and the
Lagrange multipliers are calculated through a regional linear programming model of land
allocation setup with the mathematical optimization software GAMs.
Farmers are assumed to operate under constant returns to scale (  i   j    1 ) and
subject to an elasticity of input substitution (σ) of 0.25 and a supply-land elasticity 0.7 for all
crops. An explicit exercise that shows how all the parameters are calculated may be seen in
Maneta et al. 2009 and on Torres et al. 2012.
Regional Net-Revenue Model
Once the crop and farmer specific parameters are calculated, their values are re-introduced
in (1) and a non-linear regional annual net-revenue maximization problem is set as:
max net  [ pig q̂igr ( X hig )  p jg q̂ irjg ( X hjg ) 
X hig , X hjg
g
i, j
( p
hig
 X
 X
X hig  phjg X hjg )  (ˆ ig e ig landig  ˆ jg e jg land jg )]
ˆ
ˆ
(5)
h
6
The regional model chooses X hig and X hjg such that the regional annual net-income is
maximized. qˆigr ( X hig ) and qˆigir ( X hjg ) are respectively the rainfed and irrigation production
functions, specified in (2) and (3), and parameterized with b̂hig , b̂hjg , Âig , Âjg , ̂ ig , ̂ jg , ˆ ig ,
ˆ jg .
This regional maximization problem is subject to the following set of constraints.
Resource constraints
(X
landig
 X landjg )  blandg
(6)
i, j
(X
flaborig
 X flaborjg)  b flaborg
(7)
i, j
m
m
X awjg  ( X swjg
 X pjg
)
(8)
m
m
X pjg
 Pjgam
X
m
swjg
j ,m
(9)
m
  bswg
(10)
m
Water Stress Constraint
X awjg
X landjg
 k * wuse jg
(11)
Constraints (6) and (7) establish that the total annual amount of land and family labor that
farmer g can use in the production of crops i and j throughout the agricultural year must be
less or equal to the annual total amount of land and family labor available, Bland and B flabor
respectively. Constraint (8) shows that the total amount of water used by crop j throughout
the agricultural year ( X awjg ) must be equal to the annual amount of surface water farmer g
m
) , plus the total annual amount of water used from the
decides to apply  ( X swjg
m
precipitation that falls over the land area where crop j is grown,
( X
m
pjg
) . m refers to the
m
12 months of the year. Constraint (10) says that the annual amount of surface water farmer g
m
can use to irrigate all irrigated crops,  X swjg
, must be less than or equal to the annual
amount of surface water available,
b
j ,m
m
swg
. Lastly, constraint (11) establishes that the annual
m
ratio of applied water to a hectare of land cannot fall below a certain threshold, k * wuse jg ,
where k is a parameter ranging from 0% to a 100% and wuse jg is the applied water to
hectare of land ratio normally used by farmer g on crop j. In this study k is assumed to be
0.85 . Essentially, this constraint puts an upper limit on the amount of water stress that can
be applied to a given crop.
Hydrological Model
The Buriti Vermelho River has five small reservoirs, two of which (reservoirs #2 and #5) are
shown in figure 1, and are used for irrigation by the small farmers. Each of the two
7
reservoirs has one channel that serves different parts of the watershed. Farmers that capture
water from the sub-watershed are represented by a black dot and the center pivot – CP3 in
Figure 1 below. The upstream channel (thicker red line) that comes from the mouth of the
Buriti Vermelho creek that splits into two channels by the time it reaches the farmers.
Farmers on Channel 1a and on Channel 1b –get 60% and 40% of the diverted water
respectively. Farmers on Channel 2 get surface water from the Midstream Reservoir 2 and
the center pivot (CP3) gets water from the downstream Reservoir 5. Center pivots 1 and 2
(CP 1 and CP2) are located within the basin, but withdraw water from outside the watershed
so they are not considered in the model. In terms of access to surface water, farmers
withdraw water in a cascading manner. Farmers at channels 1a and 1b get water first from a
reservoir near the creek mouth. After the diversion, the water flow that remains in the creek
goes to reservoir 2 that supplies water to the farmers at channel 2. The remaining water
flows to reservoir number 5 which then is used to irrigate the center pivot.
A hydrologic model was developed to calculate water flows in the channels. Since
the channels are not monitored, the estimation was performed by simulating discharge as a
function of the water height above the pipeline. The hydrologic model is based on
Thornthwaite and Mather, 1955, and uses the procedure developed in Liebe et al., 2009.
Simulations were done daily and the results were aggregated monthly to populate the
economic model. More specifically, the hydrologic model provides estimates of the amount
of surface water from the creek that is available monthly to each farm in the community and
m
the owner of the center pivot. That is, the hydrologic model provides estimates of bswg
in
equation (10). It also provides estimates of daily millimeters of precipitation which is
aggregated to a monthly estimate which are then used as the basis for the calculation of the
amount of precipitation the falls onto the crop j and i by farmer g.
Figure 1 – Buriti Vermelho sub-watershed
8
Channel 1a
Channel 1b
•• ••
••• •
Channel 2
Buriti Vermelho
Creek Mouth
SITE OF STUDY AND DATA
The area of study, the Buriti Vermelho sub watershed as shown in figure 1, is located about
100 kms from Brasília, Brazil. The primary data were collected in situ through a survey of
all 25 famers located in the basin who used water from the basin during the agronomic year
2007/2008 (October 2007 through September 2008). The survey was administered in two
phases: one immediately after the wet season (October 2007 – March 2008) and another
immediately after the dry season (April – September of 2008). For each farmer and crop
produced during the agronomic year, data was collected on outputs (prices received and
quantity produced) and inputs (prices paid and quantity used). The amount of surface water
m
used monthly by crop and by farmer, X swjg
, was estimated with information collected on the
frequency and duration of irrigation, the type of irrigation technology used and the type of
pump by crop and farmer. A planting and harvesting calendar for each crop and each farmer
plus the area allocated to each crop was then used to estimate the monthly precipitation that
fell on the crops, Pjgam and Pigam .
Tables 1and 2 below show the monthly surface water demand as a percentage of total
surface water supply and farmers’ descriptive statistics according to their relative position
across the basin. Considering the baseline year as a typical year, Table 1 shows that in
general farmers use a small percentage of the total surface water available (20% on average
for farmers at the channels and 16% for the center pivot ). Farmers at Channel 1b face the
highest percentages and as a consequence are closer to a surface water binding situation.
Table 2 shows that Farmers at channel 1a have on average higher annual net revenues than
farmers on channel 1b. Farmers at channel 2 are the lowest income farmers on average and
the center-pivot farmer (CP3) earns the highest net-income. Farmers at the channels use on
9
average 3 to 4 hectares of cropping land yearly, while the center pivot uses 44 hectares.2
Farmers at Channels 1a and 1b have higher net-revenue per hectare, a slightly higher
reliance on rainfed crops and have a much higher rate of water per irrigated land than
farmers at Channel 2. Farmers at the channels plant vegetables and fruits and are more
diversified than the center pivot farmer who mostly relies on grains (corn, beans, wheat and
soybeans). Lastly, although most of the land is irrigated, at least at some point in time, for
the farmers at the channels, 25% of the total water used (precipitation and surface water) is
supplemental. This percentage increases to 64% in the case of the center pivot.
Table 1 – Monthly percentage of surface water demand over monthly total water
supply at the baseline year Conditions
Months
Channel 1a
Channel 1b
Channel 2
Center Pivot
January
12.4
35.5
12.1
0.0
February
21.8
40.4
19.4
0.0
March
28.3
24.7
17.8
25.4
April
28.1
28.1
25.4
16.9
May
19.2
24.2
22.4
18.9
June
5.7
23.7
24.8
18.1
July
7.0
16.4
22.9
9.4
August
9.4
16.5
11.7
10.5
September
7.9
17.3
12.4
0.0
October
20.7
5.6
24.9
15.8
November
23.3
9.3
19.9
14.8
December
16.1
28.3
18.2
0.0
Table 2 – Descriptive statistics across basins
Farmers
Channel 1a
Channel 1b
Channel 2
CP3
Annual Average
Net-Revenue
Average
Cropping Land
Annual Water Use
per Irrigated Land
Irrigated
Land
(Brazilian Reais)
(hectares)
(m3 ̸ hectare)
(%)
30,036.00
22,635.00
15,171.00
432,208.00
3.22
4.10
3.40
44.00
Average Annual
Net-Revenue per
Hectare
(BrazilianReais)
9,340.00
5,518.00
4,459.00
2,098.00
1722
2066
1097
4412
85
70
94
62
Target Crops
Vegetables and Fruits
Vegetables and Fruits
Vegetables and Fruits
Grains
Supplemental
water use (a)
(%)
24
29
22
64
Farmers
Channel 1a
Channel 1b
Channel 2
CP3
# of Farmers
7
5
10
1
# of
crops
18
11
20
4
(a) Percentage of surface water from the reservoirs used to irrigation over the total amount of water
used (direct precipitation over the crops + surface water from the reservoirs).
2
After planting and harvesting a given crop, the plot of land may be re-used for cropping more than once in a
single agricultural year.
10
CALIBRATION, VALIDATION AND SIMULATION
Calibration and Validation
The basic model used for calibration is the one formed by equations 5 through 11 and called
Precipitation Model (MP). By calibration it is meant that the optimized output and input mix
for each farmer g, resulting from the net revenue maximization problem, is sufficiently close
to the values observed in the base year. That is within the range of plus or minus 5%
compared to the observed base year values.
Besides the fact that the basic model calibrates closely to the base year data, other
important features of the model are important to validate the model´s prediction capability.
An agricultural production model that includes precipitation and stored surface water as
inputs should show that progressive cuts in precipitation are ceteris-paribus translated into
more surface water pumping. Also it should show that increasing cuts in surface water
availability, given the level of precipitation, are translated into increasing net-revenue losses.
Effects across the basin should also be heterogeneous as farmers´ reliability on water and
access vary across the basin. These results, based on the basic MP model, are corroborated
by Figures 2, 3 and 4 below.
Figure 2 shows that cuts in precipitation from 5 to 90% will induce farmers to
substitute surface water for precipitation by increasing the amount of pumped surface water
up to 70%, given surface water costs and volume are kept at the baseline year levels. This
farmer response to dry conditions is common the world over, but is sometimes not formally
modeled, or is modeled with the two water sources being perfectly and costlessly
substituted.
Changes in Surface Water Use
%
Figure 2 – Impacts on Surface Water Use
Due to Cuts in Precipitation
80
60
40
20
0
10
20
30
40
50
60
70
80
90
Cuts in Precipitation Level
%
In terms of the impacts of cuts in surface water availability on the regional net-revenue,
Figure 3 shows that given different levels of precipitation, the higher the cut in precipitation,
the higher is the impact on net-revenue of a given cut in surface water availability. That is,
the shadow price of surface water increases with increased cuts in precipitation. At the
baseline precipitation level, impacts on net revenue start to become apparent only at 50% or
higher cuts in surface water. For reductions in precipitation by 30% and 50%, however,
impacts on net-revenue starts at much lower cuts in surface water. Similar patterns can be
seen in Figure 3 with successive cuts in surface water availability.
11
Decreases in Net Revnues
%
Figure 3 – Impacts on regional net revenue due to cuts in surface
water availability
0
-10
-20
-30
-40
-50
-60
-70
-80
Base Precipitation
30% Cut in Precipitation
50% Cut in Precipitation
10 20 30 40 50 60 70 80 90
Cuts in Surface Water Volume
%
Across the basin, drought impacts on water supply and net revenue are shown to be
heterogeneous. Assuming a 50% and a 30% cut in precipitation, the hydrological model
estimates the corresponding impacts of these precipitation cuts on the reservoir levels (Table
3). As expected, a 30% cut in precipitation will cause a lower impact on surface water
supply than a 50% cut but these impacts are not homogenous. Farmers at the end of the
stream are hit harder successively in terms of access to water in the reservoirs. Figure 4
below shows, for example, that farmers at Channel 1b face a higher impact compared to
farmers at Channel 1a. This is likely because they have less access to water due to their
channel having a lower conveyance capacity. Table 3 shows that farmers at Channel 1a are
much more effective in compensating for the loss in precipitation by pumping more surface
water then farmers at channel 1b. The relatively higher reliance on purely rainfed crops by
farmers at Channel 1b might also help in explaining this pattern (Table 2). The unfavorable
position of farmers at Channel 2 and the center pivot 3 at the middle and end of the stream
explains, in great part, the result that they are hit hardest in terms of water supply for both
the 30 or 50% cuts in precipitation (Table 3). All farmers compensate for the 30% cut in
precipitation by supplementing with more surface water. At the 50% cut, farmers at the
channels repeat the supplemental water use pattern, in some cases, more than doubling
amount of supplemental water use (Channel 1b), but the center-pivot farmer ends up
reducing the amount of supplemental water used by 34%. This is likely due to the fact that
water supply becomes more strongly binding since it drops 92% in volume.
Table 3 – Changes in surface water supply and surface water use
across the basin after the 30 and 50% precipitation cuts
Farmers
Channel 1a
Channel 1b
Channel 2
CP3
30%
Change
Surface
Supply
(%)
-26
-33
-54
-75
in Change
Water Surface
Use
(%)
93
48
67
17
50%
in Change
in
Water Surface Water
Supply
(%)
-46
-49
-71
-92
Change
Surface
Use
(%)
132
44
41
-34
in
Water
12
Impacts on farmers’ revenue from cuts in precipitation and corresponding changes in the
surface water supply estimated by the hydrological model may be seen in Figure 4 below. In
general, an increase in drought severity from having a 30% and a 50% cut in precipitation
causes an average decrease in expected revenue of 14% and a 23% respectively. As
expected, the cost of cuts increases nonlinearly with the severity of drought. Also on
average, costs due to a 50% cut in precipitation are 64% higher than for the 30% cut
scenario.
The farmers’ position across the basin, access to surface water and reliance on
irrigation or rainfed crops explain the range of drought impacts on net-revenue. Farmers at
Channel 1b suffer more than farmers at Channel 1a either at a 30 or 50% cut in precipitation
(see Fig. 4 below) since Channel 1b has a lower capacity (in fact Table 2 shows that water
supply drops relatively more for farmers at Channel 1a than on Channel 1b) and its farmers
rely more on rainfed crops than the ones on Channel 1a. Relatively to farmers at Channels
1a and 1b, farmers at Channel 2 suffers more despite their relatively more extensive use of
irrigation (94% of their land is irrigated) and lower rate of irrigation water per hectare on
irrigated land (Table 1). Their downstream position and relatively higher decrease in surface
water supply (-54 and -71% ) due to the cuts in precipitation likely explain their net-revenue
patterns. The Center pivot relies more on supplemental water, but its water supply decreases,
relatively to other farmers, the most (75 and 92% , Table 2) due to its downstream position.
As a consequence the drop in its net revenue is the highest. Also the fact that the center
pivot farmer has a relatively high percentage of rainfed crops may also help to explain the
pattern of drought impacts on its net-revenue.
Figure 4 – Drought impacts on net revenue across the basin
0.0
Change in Net-Revenue
%
-5.0
-10.0
-15.0
50% Cut in Precipitation
-20.0
30% Cut in Precipitation
-25.0
-30.0
-35.0
-40.0
Channel-1a Channel-1b
Channel2 Center Pivot
Simulation
Besides the Precipitation Model (MP) as defined by equations 5 through 11 we construct an
alternative model (called hereafter Model with Pooled Constraint or MPC) in which the
maximization is subject to a simpler set of constraints. The MPC uses the same equation 5
and constraints (6), (7), (8) and (11) as used in the MP model, but constraints (9) and (10)
are substituted by a pooled water constraint such as
X
j ,m
m
swjg
m
m
  X pjg
  bswg
  Pjgam
j ,m
m
(12)
j ,m
13
Constraint (12) states that for each farmer g the total annual amount of water applied to
m
m
irrigated crops,  X swjg
, must be less or equal to the annual amount of surface
  X pjg
j ,m
water available (  b
m
swg
(  Pjgam ).
j ,m
) plus the total annual amount of rain that falls onto irrigated crops
m
j ,m
In the MP model, substitution between precipitation and surface water is explicitly
formalized and restricted to reflect the distribution of precipitation. In the MPC model,
precipitation and irrigation water are aggregated together in a single water supply amount as
is generally done in the models applied to arid and semi-arid regions cited in the
introduction. For both of these versions two drought scenarios are simulated: a 30% and a
50% reduction in precipitation over all months of the year.
RESULTS
Pooling Versus Non-Pooling
Figure 5 below shows the net-revenue impacts under a 30% cut in precipitation predicted by
the precipitation basic model (MP), in red bars, and the model with pooled water constraint
(MPC) in grey bars. In here as in Table 3, we use the hydrological model estimates of the
corresponding impacts of a 30% cut in precipitation on reservoir levels. In general, by
showing a larger net-revenue across the basin, the traditional MPC pooled model
underestimates the expected net-revenue losses compared to the more accurately specified
MP precipitation model. This underestimation pattern is more evident for farmers at
Channel 2 and the Center Pivot (CP3). In terms of the average impacts on regional
production, the MPC model may underestimate by as much as 53% at the baseline
precipitation and surface water supply conditions and by 22% under the 30% precipitation
cut.
Figure 5. Net revenue under the effects of a 30% cut in
precipitation. MP and MPC models
Net Revenue
(1000's)
500
400
300
200
100
0
Stochastic MP and MPC models.
Stochastic versions of the PM and MPC model were developed using the synthetic
hydrologic precipitation records for the region to generate a series of stochastic precipitation
14
values. We fitted a log-normal distribution for the precipitation with parameters of 4.6 and
0.5 as the mean and standard deviation respectively. This same distribution is used to
generate a series of stochastic surface water supply values. 3 The purpose in this section is
to use both model versions as a tool to perform a risk analysis based on the variability of net
returns and on the use of supplemental surface water to mitigate short run drought effects.
The two models were solved for each of the 50 precipitation and surface water supply
realizations and the results were used to calculate the difference in the revenue risk between
the two model specifications.
Table 4: Average net income variation by farm size(a)
Average Net Income per Hectare
Large
Medium
Small
Precipitation Model
Mean
Standard deviation
Coefficient of Variation
Percent Supplemental
2124.3
297.3
0.14
64.9
8885.7
8952.2
1.01
29.8
5631.0
8510.6
1.51
25.4
Pooled Model
Mean
Standard deviation
Coefficient of Variation
Percent Supplemental
2970.1
381.7
0.13
0
10695.9
8702.0
0.81
0
7244.5
8611.8
1.19
0
(a) Large, one farm with 206 hectares of annual area under plow; medium,
7 farms with 4 to 7 hectares of land area under plow; small, 15 farms
with area under plow below 4 hectares.
Table 4 shows the difference in farm income risk measured by the two model specifications.
Not only does the pooled model underestimate the expected loss due to stochastic rainfall
but for all farm sizes, the risk as summarized by the coefficient of variation is also
downplayed by the water pooling model. Under both model specifications, net income risk
is reduced by larger farm sizes and a greater proportion of supplemental water used by the
farmers. For each farm, there are 50 realization values of net-revenue per hectare. The mean
values and standard deviation are calculated considering all farms within each type (large,
medium and small) over all 50 realizations. Percent supplemental is the average percentage
of surface water used over total applied water (surface water + precipitation), considering all
farms within each type (large, medium and small) over all 50 realizations. Percent
supplemental in the MPC model is zero since surface water is pooled with precipitation,
which is costless.
The results suggest that the variability of net returns decreases with increased farm
size and that supplemental water use reduces income variability. Moving from large to
medium farms, the coefficient of variation sharply increases by seven and six fold in the MP
and MPC models respectively. The difference is even larger if we compare large with small
farms. From medium to small, the coefficient of variation increases around 50% in both
versions. At the same time (based on the MP model) the percentage of supplemental water
3
Ideally, for a given distribution assumed for precipitation, the hydrological model should generate the
corresponding distribution of values of the surface water volume in the reservoirs. Although it is likely that the
water volume follows the same distribution assumed for precipitation, it is not likely they share the same
parameters (mean and standard deviation). We assume here, however, they share the same parameters.
15
used decreases by half from large to medium farms and by 15% from medium to small. 4
The coefficient of variation estimates for the pooled MPC model are consistently lower
compared to the MP model values. In addition, the underestimation gets more evident for
the cases of medium and small farmers. In general, Table 4 shows that 1) revenue risk is
greater for small farmers, 2) supplementary irrigation reduces risk, and 3) pooling water
sources in a single and undistinguishable source results in underestimation of the revenue
risk..
Frequency
Figure 6: Stochastic Precipitation-Net Income Distribution- Small Farms- PM and
MPC Models
100
90
80
70
60
50
40
30
20
10
0
MP Model (non-pooled)
MPC Model (pooled)
Net Revenue per Hectare
Figure 6 shows the frequency distribution of net income for small farms as generated by the
MP and MPC model specifications. From the deterministic results we know that the mean
income is higher for the MPC model. In addition figure 6 shows that the income distribution
under the pooled MPC model has a positive skew on the upside. This skewedness reflects
the ability of optimizing farmers to make better use of the total water resources when they
are plentiful if they are, unrealistically, assumed to be completely flexible and substitutable
across crops and growing seasons.
CONCLUSION
This paper shows that for many irrigation systems that provide supplemental water to
seasonal precipitation, the specification of substitution between irrigation and precipitation
has a significant impact on model results. In short, models that assume a fully substitutable
pooled water supply underestimate the net income cost of drought and the value of
supplemental water, in addition, the pooled model introduces a positive upward bias in net
income. In areas where the water access rule is a first-come, first-serve basis and farmers are
located in a cascading series along a river with farmers situated downstream are more likely
to be inflicted with the greatest cuts in supply. For a given drought situation, they get the
remaining water after it has been diverted to satisfy the demands of farmers located
upstream. This paper shows this effect using a fully integrated modeling approach that
combines hydrology and economics and can be used for the estimation and quantification of
the drought impacts on agricultural income. It also shows that the use of modeling
approaches that do not explicitly consider the substitution relationship between surface
water and precipitation will underestimate the average drought impacts and the risk implied
by stochastic precipitation. While the current empirical example is too small to generalize, it
4
We should bear in mind that medium and small farmers are more comparable because they share similar
irrigation technologies and product mixes (vegetables and fruits). The large farm operation in the sample uses
center-pivot technology and grows grains.
16
does show that modelling situations, where irrigation is supplemental to precipitation, using
conventional arid region pooled water models will introduce bias into both the estimates of
expected net revenue and their variance. Clearly, the importance of this bias will depend on
the relative importance of expected precipitation to the total seasonal water requirement. In
many irrigated systems the contribution of precipitation is substantial, and with it the
potential for bias from incorrectly specified models.
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